Theorem eltrrels2 38914

Description: Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)

Assertion

Ref	Expression
eltrrels2	⊢ (𝑅 ∈ TrRels ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))

Proof of Theorem eltrrels2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftrrels2 38910	. 2 ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}
2		coideq 38499	. . 3 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑅))
3		id 22	. . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
4	2, 3	sseq12d 3969	. 2 ⊢ (𝑟 = 𝑅 → ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅))
5	1, 4	rabeqel 38508	1 ⊢ (𝑅 ∈ TrRels ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903   ∘ ccom 5636   Rels crels 38436   TrRels ctrrels 38448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-rels 38691  df-ssr 38829  df-trs 38907  df-trrels 38908

This theorem is referenced by:  eltrrelsrel  38916